Abstract-The concept of pseudospectra was introduced by
Trefethen during the 1990s and became a popular tool to
explain the behavior of non-normal matrices. It is well known
that the zeros of a polynomial are equal to the eigenvalues of
the associated companion matrix. It is feasible to do the
sensitivity analysis of the zeros of polynomials by the tool of
pseudospectra of companion matrices. Thus, the pseudospectra
problem of companion matrices is meaningful. In this paper,
we propose a fast algorithm for computing the pseudospectra
of companion matrices by using fast QR factorization. At last,
numerical experiments and comparisons are given to
illustrate(confirm) the efficiency of the new algorithm.
Keywords-Pseudospectra; Companion matrix; Grid; SVD; QR
factorization; Field of values; Gershgorin disk

I. INTRODUCTION
The concept of pseudospectra was introduced by
Trefethen to explain the behavior of non-normal matrices [1,
2, 3, 4]. The pseudospectra of a square matrix are the set of all
eigenvalues of complex matrices within a given distance. It is
a useful tool for understanding the behavior of various matrix
processes. Many phenomena (for example, hydrodynamic
instability and convergence of iterative methods for linear
systems) can not be accounted for by eigenvalue analysis but
are more understandable by examining the pseudospectra [5,
6, 7, 8].
There are the following equivalent definitions of
pseudospectra. Let · be a matrix norm induced by a vector
norm.
（1） ( ) ( )
{ }
-1
-1
= : A z C zI A
c
c A e ÷ > ;
（2）
( ) = A
c
A ( ) { }
: , z C z A E E E c e e + s f or some wi t h

（4）If · is the the 2-norm, the following
definition is also equivalent:
( ) ( ) { }
min
= : - A z C zI A
c
o c A e s ;
Note that if - zI A is singular, we
denote ( )
-1
- = A zI · and ( )
min
o · denotes the minimum
singular value. From these definitions, it immediately follows
that for any
1 2
> c c ,
( ) ( )
1 2
A A
c c
A _A .Furthermore, ( )
0
A A
coincides with the eigenvalues of A.Especially, if matrix Ais
normal, the ( ) A
c
A is just the closed " c -neighborhood
of ( ) A A .
The popular computational method for matrix
pseudospectra are the grid method based on basic SVD
(denoted Grid-SVD algorithm). Also, there are other
computational methods, such as inverse iteration, Lanczos
iteration, Arnoldi iteration, continuation and so on [2, 3, 4, 5,
6]. And, pseudospectra of rectangular matrices have been
considered by Wright and Trefethen [11].

polynomial ( ) p z are equal to the eigenvalues of the
associated companion matrix A.
In this paper, we propose a fast algorithm for computing
the pseudospectra of companion matrices by using
fast QR factorization in section 2. Numerical experiments and
comparisons are given to illustrate the efficiency of the new
algorithm in section 3. Conclusion and remarks are given in
section 4.

Based on a theory of numerical linear dependence we get a
new definition of pseudospectra. For any given z C e and
each 0 c > , let | |
1 2
= , , ,
n
B zI A | | | = ÷ . The
pseudospectra of matrix Acan be defined by
( )
{ } 1 2 -1
= , , ,
n n
A z C E
c
| | | | c A e ( s
¸ ¸
(5)
Before we present a new algorithm for computing
pseudospectra of companion matrix, we need to introduce the
fast QR factorization of upper-Hessenberg matrix. The cost of
the fast QR factorization of upper-Hessenberg matrix using
Givens rotations is
( )
2
O n operations because of its upper-
Hessenberg structure while it is
( )
3
O n operations
for QR factorization of general matrix. And, the cost of the
singular value decomposition (SVD) of general matrix is also
( )
3
O n operations. Therefore, we can present a fast algorithm
for computing pseudospectra of companion matrix which
needs much less cost and is much faster and than the
traditional algorithm (Grid SVD).
Because a companion is an upper-Hessenberg matrix, we
consider the QR factorization of upper Hessenberg matrices
using Givens rotations. In detail, given a 2 2 × matrix
11 12
21 22
=
a a
A
a a
(
(
¸ ¸

III. NUMERICAL EXPERIMENTS
In this section, we give some numerical experiments and
comparisons to confirm the results in this paper. Here, all the
computations are finished with MATLAB 6.5 on PC (Intel(R)
Pentium(R), Processor 1500MHz 1.50GHz, Memory 256MB).

IV. CONCLUDING REMARKS
Perhaps pseudospectra will play a role tool in breaking
down walls between the theorists of functional analysis and
the engineers of scientific computing. Computing
pseudospectra will be a routine matter among scientists and
engineers who deal with non-normal matrices [1]. Thus, we do
the interesting research work on the pseudospectra of
matrices.
In this paper, we propose a fast efficient algorithm for
computing the pseudospectra of companion. And we discuss
the approach for computing the pseudospectra region easily.
Furthermore, we discuss the relationship between pseudozeros
sets of polynomial and pseudospectra of the associated
companion matrix. The numerical experiments and
comparisons are given to confirm the results in this paper.
In fact, the idea in this paper can be generalized to the
pseudospectra of rectangular matrices and other structure
matrices. Furthermore, the idea in this paper may provide some
insights for the pesudospectra in other norms, i.e. weighted
pseudospectra.
ACKNOWLEDGMENT
This work was supported by the Fundamental Research
Funds for the Central Universities (No.NZ2012307).

Zhengsheng Wang. Born in 1972, he received the Bachelor degree,
Master degree and Ph.D. of Computational Mathematics in
Engineering in Nanjing University of Aeronautics and Astronautics
in China. He has a academic access Swinburne University of
Technology in Australia.
He is an associate professor of Computational Mathematics in
Nanjing University of Aeronautics and Astronautics. Since 2000, he
has involved in the completion of five projects with national,
provincial or departmental level. And now he has three projects for
researching. He has published over 20 academic papers including 8
for SCI, EI or ISTP retrieval.

Chuntao Liu. Born in 1987, she received the Bachelor of
Mathematics in Fuyang Normal College in China; and now studying
in Nanjing University of Aeronautics and Astronautics as a graduate
of Computational Mathematics.